Matchings extend to Hamiltonian cycles in hypercubes | Open Problem Garden. \begin{question} Does every \Def[matching]{matching} of \Def[hypercube]{hypercube} extend to a \Def[Hamiltonian cycle]{Hamiltonian path}? \end{question} This question is due to Ruskey and Savage and appears in [RS] (page 19, question 3). The answer is positive for $d$-cube if $d \le 4$. Fink [F] proved Kreweras' conjecture [K] which asserts that every \Def[perfect matching]{matching} of hypercube extends to a Hamiltonian cycle.
[K] G. . % Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. . * indicates original appearance(s) of problem. Graceful Tree Conjecture | Open Problem Garden. \begin{conjecture} All trees are graceful \end{conjecture} Label the vertices of a simple undirected graph $G(V,E)$ (where $|V| = n$ and $|E| = m$) with integers from $0$ to $m$. Now label each edge with absolute difference of the labels of its incident vertices. The labeling is said to be graceful if the edges are labelled $1$ through $m$ inclusive (with no number repeated).
A graph is called graceful if it has at least one such labeling. This labeling was originally introduced in 1967 by Rosa. The name graceful labeling was coined later by Golomb. Gracefully labeled graphs serve as models in a wide range of applications including coding theory and communication network addressing. The graceful labeling problem is to determine which graphs are graceful. Despite numerous (more than 200) publications on graceful labeling for over three decades, only a very restricted classes of trees (and also of some other graphs) have been shown to be graceful. Reconstruction conjecture | Open Problem Garden. Bigraph. Anatomy of a bigraph[edit] Aside from nodes and (hyper-)edges, a bigraph may have associated with it one or more regions which are roots in the place forest, and zero or more holes in the place graph, into which other bigraph regions may be inserted.
Similarly, to nodes we may assign controls that define identities and an arity (the number of ports for a given node to which link-graph edges may connect). These controls are drawn from a bigraph signature. In the link graph we define inner and outer names, which define the connection points at which coincident names may be fused to form a single link. Foundations[edit] A bigraph is a 5-tuple: where is a set of nodes, is a set of edges, is the control map that assigns controls to nodes, is the parent map that defines the nesting of nodes, and is the link map that defines the link structure. The notation indicates that the bigraph has holes (sites) and a set of inner names and regions, with a set of outer names .
See also[edit] Bibliography[edit] Matchings extend to Hamiltonian cycles in hypercubes | Open Problem Garden. Spectral decision diagrams using graph transformations. BibTeX @INPROCEEDINGS{Thornton01spectraldecision, author = {Mitchell Thornton and Rolf Drechsler}, title = {Spectral decision diagrams using graph transformations}, booktitle = {in Design Automation and Test in Europe. (Munich}, year = {2001}, pages = {713--717}} Bookmark OpenURL Abstract Spectral techniques are powerful methods for synthesis and verification of digital circuits. Citations. Graph Theory: Electronic preview. Graph Theory Tutorials. Chris K. Caldwell (C) 1995 This is the home page for a series of short interactive tutorials introducing the basic concepts of graph theory. There is not a great deal of theory here, we will just teach you enough to wet your appetite for more! Most of the pages of this tutorial require that you pass a quiz before continuing to the next page.
So the system can keep track of your progress you will need to register for each of these courses by pressing the [REGISTER] button on the bottom of the first page of each tutorial. (You can use the same username and password for each tutorial, but you will need to register separately for each course.) Introduction to Graph Theory (6 pages) Starting with three motivating problems, this tutorial introduces the definition of graph along with the related terms: vertex (or node), edge (or arc), loop, degree, adjacent, path, circuit, planar, connected and component. Euler Circuits and Paths Coloring Problems (6 pages) Adjacency Matrices (Not yet available.) Cayley graphs and the algebra of groups.
This is a sequel to my previous blog post “Cayley graphs and the geometry of groups“. In that post, the concept of a Cayley graph of a group was used to place some geometry on that group . In this post, we explore a variant of that theme, in which (fragments of) a Cayley graph on is used to describe the basic algebraic structure of , and in particular on elementary word identities in . Throughout this post, we fix a single group , which is allowed to be non-abelian and/or infinite. In the previous post, we drew the entire Cayley graph of a group . Of the group , and one draws a directed edge from to labeled (or “coloured”) by the group element for any ; the graph consisting of all such vertices and edges will be denoted . Looks like this: Figure 1.
One usually does not work with the complete Cayley graph. . , in which the edge colours are restricted to a smaller subset of , such as a set of generators for . Cayley graphs are left-invariant: for any , the left translation map is a graph isomorphism. Of . . . Cayley graphs and the geometry of groups. In most undergraduate courses, groups are first introduced as a primarily algebraic concept – a set equipped with a number of algebraic operations (group multiplication, multiplicative inverse, and multiplicative identity) and obeying a number of rules of algebra (most notably the associative law).
It is only somewhat later that one learns that groups are not solely an algebraic object, but can also be equipped with the structure of a manifold (giving rise to Lie groups) or a topological space (giving rise to topological groups). (See also this post for a number of other ways to think about groups.) Another important way to enrich the structure of a group is to give it some geometry. Of the group . A generated group; in many important cases the set of generators is finite, leading to a finitely generated group. Gives rise to the word metric on , defined to be the maximal metric for which for all and (or more explicitly, is the least for which for some ) . . . , and vertices labeled by elements of to.
Graceful Tree Conjecture « My Brain is Open. Graceful Tree Conjecture (GTC) is one of my favorite open problems. I posted it on Open Problem Garden couple of years back. I enjoy reading papers related to graceful labeling and try to keep as up-to-date as possible with the progress towards GTC. Here is a brief introduction to GTC. Graceful Labeling : Label the vertices of a simple undirected graph (where and ) with integers from to . Now label each edge with absolute difference of the labels of its incident vertices. A graph is called graceful if it has at least one such labeling. Gracefully labeled graphs have applications in coding theory and communication network addressing. Graceful Tree Conjecture (GTC) : All trees are graceful.Ringel’s Conjecture : If is a fixed tree with edges, then complete graph on vertices decomposes into copies of . A caterpillar graph is a tree such that if all leaves and their incident edges are removed, the remainder of the graph forms a path.
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